The aim of this course is to introduce basic notions of commutative algebra, a discipline that flourished after the work of Hilbert and Dedekind at the end of the 19-th century, and developed during the 20-th century. Commutative algebra is largely concerned with the study of certain rings, called Noetherian, in which every ideal is finitely generated. The development of commutative algebra is often motivated by natural and simple-looking questions. Among them: what kind of factorization is available for Noetherian rings? What is the "correct" notion of dimension for a ring?
The aim of the course is to introduce some basic notions in commutative algebra such as localization, tensor product, Noetherian and Artinian modules, Krull dimension and integral dependence.
The aims of the course are:
1) To introduce the student to some techniques which are typical of of commutative algebra such as localization, tensor product and extension/contraction of ideals along morphisms.
2) To introduce Notherian and Artinian modules, and to develop the theory of associated primes of a module in relation to primary decomposition (a.k.a. the "substitute" of unique factorization) for ideals in Noetherian rings.
3) To characterize Artinian rings by introducing the length of a module and the Krull dimension of a ring; moreover, using such a characterization, the goal is to prove Krull's Height Theorem.
4) To introduce the concept of integral dependence in order to compute the Krull dimension of algebras which are finitely generated over a field.
The expected learning outcomes are:
1) A strengthening in the knowledge of basic algebraic notions thanks to the theoretical aspects developed in class and thanks to the discussion of assigned exercises; a good handling of standard techniques in commutative algebra.
2) At the end of Algebra Commutativa 1, the student has learned the features of Noetherian rings, knows the characteristics of primary decompositions of ideals, and knows how to compute them in some notable cases (e.g. monomial ideals).
3) At the end of Algebra Commutativa 1 the student has learned what Krull dimension is, and knows ways to compute it in the case of local rings and in the case of algebras which are finitely generated over a field.
The student is expected to be familiar with groups, rings, ideals, modules, algebras, quotients and homomorphisms.
Lessons will be in presence. Most of the available hours will be devoted to the development of the theoretical part of the course; exercise sheets will be made available during the semester, and will be discussed collectively in the remaining hours.
1) Rings, Ideal, modules and algebras. Rings and modules of fractions.
2) Chain conditions, Noetherian and Artinian rings.
3) Primary decomposition, associated primes.
4) Tensor products of modules, tensor algebra, symmetric algebra and exteriro algebra.
5) Rings of invariants, Reynolds operator, pure subrings, direct summands and algebra retracts.
6) Dimension theory. Krullhauptidealssatz and its variations. System of parameters and dimension of local rings.
7) Integral dependence, Noether normalization and dimension of finitely generated K-algebras.
- Notes in Italian, made available on Aulaweb
- D. Eisenbud, Commutative Algebra with a View toward algebraic geometry, Springer 1994
- Atiyah MacDonald, Introduction to Commutative Algebra, Addison-Weysley 1969.
Ricevimento: By appointment
ALESSANDRO DE STEFANI (President)
ALESSIO CAMINATA
ALDO CONCA (President Substitute)
EMANUELA DE NEGRI (President Substitute)
Accordingly with the academic calendar.
The examination is oral.
The student will be evaluated on the theoretical aspects developed during the lectures and on the resolution of some exercises, typically along the lines of those discussed in class.
The assessment will be based both on the knowledge of the topics and on the ability to present them in a formal, coincise and correct way.
The grade is based on the performance at the oral exam, and participation during the activities (in class lectures, exercise sessions) of the semester.
Attendance in person is highly recommended.
Students with DSA certification ("specific learning disabilities"), disability or other special educational needs must follow the instructions available on Aulaweb at https://2023.aulaweb.unige.it/course/view.php?id=12490#section-3 in order to agree on special arrangements.
Requests should be sent well in advance (at least 10 days) before the date of the exam by sending an email to the professor with the "Referente di Scuola" and the DSA office in Cc (please see instructions).
